Met dank aan Rita Biesemans :

Names of the Jewish Months
The Lunar Cycle
The Jewish calendar is based on lunar cycles.  1 Towards the beginning of the moon’s cycle, it appears as a thin crescent. That is the signal for a new Jewish month. The moon grows until it is full, the middle of the month, and then it begins to wane until it cannot be seen. It remains invisible for approximately two days2—and then the thin crescent reappears, and the cycle begins again.
The entire cycle takes approximately 29½ days.3 Since a month needs to consist of complete days, a month is sometimes twenty-nine days long (such a month is known as chaser, “missing”), and sometimes thirty (malei, “full”).

Knowing exactly when the month begins has always been important in Jewish practice, because the Torah schedules the Jewish festivals according to the days of the month.
The first day of the month, as well as the thirtieth day of a malei month, is called Rosh Chodesh, the “Head of the Month,” and has semi-festive status. See Why is Rosh Chodesh sometimes one day and sometimes two?
The Jewish Months
Nissan is the first month on the Jewish calendar. Before the Jews left Egypt, on the first day of the month of Nissan, G‑d told Moses and Aaron: “This chodesh (new moon, or month) shall be to you the head of months.”4 Thus the peculiarity of the Jewish calendar: the year begins on Rosh Hashanah, the first day of the month of Tishrei (the anniversary of the creation of Adam and Eve), but Tishrei is not the first month. Rosh Hashanah is actually referred to in the Torah as “the first day of the seventh month.”5
The Jewish Months and their Special Dates
Jewish Month Approximate Secular Date
This Month’s Special Dates

Nissan
March–April
Passover

Iyar
April–May
Lag B’Omer

Sivan
May–June
Shavuot

Tammuz
June–July
 
Menachem Av
July–August
Tisha B’AV

Elul

August–September

 
Tishrei
September–October
The High Holidays (Rosh Hashanah and Yom Kippur), Sukkot, Shmini Atzeret, and Simchat Torah

Marcheshvan
October–November
 

Kislev

November–December
Chanukah

Tevet
December–January
Conclusion of Chanukah

Shevat
January–February
Tu B’Shvat

Adar
February–March
Purim